Analytic polyhedron

In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Cⁿ of the form

${\displaystyle P=\{z\in D:|f_{j}(z)|<1,\;\;1\leq j\leq N\}}$

where D is a bounded connected open subset of Cⁿ, ${\displaystyle f_{j}}$ are holomorphic on D and P is assumed to be relatively compact in D.^[1] If ${\displaystyle f_{j}}$ above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy and it is thus pseudo-convex.

The boundary of an analytic polyhedron is contained in the union of the set of hypersurfaces

${\displaystyle \sigma _{j}=\{z\in D:|f_{j}(z)|=1\},\;1\leq j\leq N.}$

An analytic polyhedron is a Weil polyhedron, or Weil domain if the intersection of any k of the above hypersurfaces has dimension no greater than 2n-k.^[2]

See also

• Behnke–Stein theorem
• Bergman–Weil formula
• Oka–Weil theorem